Solve by Factoring 1/(x-2)+1/(x^2-7x+10)=6/(x-2)

$\frac{1}{x - 2} + \frac{1}{x^{2} - 7 x + 10} = \frac{6}{x - 2}$

Step 1

Subtract $\frac{6}{x - 2}$ from both sides of the equation.

$\frac{1}{x - 2} + \frac{1}{x^{2} - 7 x + 10} - \frac{6}{x - 2} = 0$

Step 2

Simplify $\frac{1}{x - 2} + \frac{1}{x^{2} - 7 x + 10} - \frac{6}{x - 2}$ .

Combine the numerators over the common denominator.

$\frac{1 - 6}{x - 2} + \frac{1}{x^{2} - 7 x + 10} = 0$

Subtract $6$ from $1$ .

$\frac{- 5}{x - 2} + \frac{1}{x^{2} - 7 x + 10} = 0$

Simplify each term.

Move the negative in front of the fraction.

$- \frac{5}{x - 2} + \frac{1}{x^{2} - 7 x + 10} = 0$

Factor $x^{2} - 7 x + 10$ using the AC method.

Consider the form $x^{2} + b x + c$ . Find a pair of integers whose product is $c$ and whose sum is $b$ . In this case, whose product is $10$ and whose sum is $- 7$ .

$- 5, - 2$

Write the factored form using these integers.

$- \frac{5}{x - 2} + \frac{1}{(x - 5) (x - 2)} = 0$

To write $- \frac{5}{x - 2}$ as a fraction with a common denominator, multiply by $\frac{x - 5}{x - 5}$ .

$- \frac{5}{x - 2} \cdot \frac{x - 5}{x - 5} + \frac{1}{(x - 5) (x - 2)} = 0$

Write each expression with a common denominator of $(x - 2) (x - 5)$ , by multiplying each by an appropriate factor of $1$ .

Multiply $\frac{5}{x - 2}$ by $\frac{x - 5}{x - 5}$ .

$- \frac{5 (x - 5)}{(x - 2) (x - 5)} + \frac{1}{(x - 5) (x - 2)} = 0$

Reorder the factors of $(x - 5) (x - 2)$ .

$- \frac{5 (x - 5)}{(x - 2) (x - 5)} + \frac{1}{(x - 2) (x - 5)} = 0$

Combine the numerators over the common denominator.

$\frac{- 5 (x - 5) + 1}{(x - 2) (x - 5)} = 0$

Simplify the numerator.

Apply the distributive property.

$\frac{- 5 x - 5 \cdot - 5 + 1}{(x - 2) (x - 5)} = 0$

Multiply $- 5$ by $- 5$ .

$\frac{- 5 x + 25 + 1}{(x - 2) (x - 5)} = 0$

Add $25$ and $1$ .

$\frac{- 5 x + 26}{(x - 2) (x - 5)} = 0$

Factor $- 1$ out of $- 5 x$ .

$\frac{- (5 x) + 26}{(x - 2) (x - 5)} = 0$

Rewrite $26$ as $- 1 (- 26)$ .

$\frac{- (5 x) - 1 (- 26)}{(x - 2) (x - 5)} = 0$

Factor $- 1$ out of $- (5 x) - 1 (- 26)$ .

$\frac{- (5 x - 26)}{(x - 2) (x - 5)} = 0$

Rewrite $- (5 x - 26)$ as $- 1 (5 x - 26)$ .

$\frac{- 1 (5 x - 26)}{(x - 2) (x - 5)} = 0$

Move the negative in front of the fraction.

$- \frac{5 x - 26}{(x - 2) (x - 5)} = 0$

Step 3

Set the numerator equal to zero.

$5 x - 26 = 0$

Step 4

Solve the equation for $x$ .

Add $26$ to both sides of the equation.

$5 x = 26$

Divide each term in $5 x = 26$ by $5$ and simplify.

Divide each term in $5 x = 26$ by $5$ .

$\frac{5 x}{5} = \frac{26}{5}$

Simplify the left side.

Cancel the common factor of $5$ .

Cancel the common factor.

$\frac{5 x}{5} = \frac{26}{5}$

Divide $x$ by $1$ .

$x = \frac{26}{5}$

Solve by Factoring 1/(x-2)+1/(x^2-7x+10)=6/(x-2)

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